Talk:Order of operations

left to right

I am not surprised that my removing the false information "operations with the same precedence are generally performed left to right" was reverted. So many people have been taught that false "rule" in grade school that many people insist that what they learned in grade school is true. But all mathematicians know that addition is commutative and associative and multiplication is commutative and associative, and mathematicians generally perform operations in whatever order is most convenient.

It is a bit ironic that I think 12/6*2 = 4, which is what you get when you perform operations left to right. But most physicists insist that 12/6*2 = 1. Of course, my reasoning has nothing to do with left to right. It makes sense to me that subtraction is addition of the opposite and division is multiplication by the reciprocal. It is strange that after all these centuries, there is nobody who can settle the question. Rick Norwood (talk) 10:02, 5 September 2023 (UTC)

You might see from my edit summary that my reason for the revert was that your new text was flawed, too (as was/is the previous text). If you come up with a better suggestion how to fix the false information, I won't object.

As for your 2nd paragraph above, there is no question to be settled - it is very common in mathematics that different authors introduce different ("local") conventions and use them afterwards. - Jochen Burghardt (talk) 17:14, 5 September 2023 (UTC)

Agreed. Mathematics is a human language and like any other human language there are variations and no universal "correct" standard. This article presents a set of conventions that are not universally applicable as there is not a set of rules that are universally applicable.

Also agree that the current wording is flawed. Where there is a specification to be followed (e.g. computer languages, spreadsheet and other number crunching software) almost everything evaluates addition/subtraction left-to-right (with subtraction interpreted as adding the inverse)* while other non-transitive operations such as division and exponentiation are sometimes left-to-right and sometimes right-to-left. Hence all those ambiguous memes that have everybody arguing on facebook.

In short, there is no convention for evaluating expressions like 12/6*2. And we shouldn't imply that there is.

Perhaps we should say something like addition and subtraction is usually performed left-to-right but there is no general agreement for division or exponentiation. We'd need a good source to back it up, and it may be a distraction this early in the article. Or we could just remove the sentence. Not really sure what is the best approach.

• And when done this way, there's no need for a rule since you get the same result due to associativity.

Mr. Swordfish (talk) 18:31, 5 September 2023 (UTC)

Do you have a source for "But most physicists insist that 12/6*2 = 1."? 62.46.182.236 (talk) 23:00, 24 February 2024 (UTC)

Yes, very strange that this claim went unchallenged!

Unless perhaps something has changed very drastically in the decades since I studied physics that would explain such a claim? I have heard that physics is in a bad shape, and this would explain a lot. So I would be happy to be enlightened. 118.208.8.117 (talk) 04:21, 24 November 2024 (UTC)

This is a more-than-a-year-old conversation and the article has since been improved to discuss this point in much greater detail. To reiterate though, people rarely if ever write anything similar to ⁠$12/6\times 2$⁠. What they do routinely write is expressions like ⁠$a/bc$⁠, which is interpreted to mean ⁠${\tfrac {a}{bc}}$⁠. –jacobolus (t) 03:47, 12 December 2024 (UTC)

Unless otherwise stated, the default convention is left-to-right. Physics journals use a different convention to save space in inline expressions. VaiaPatta (talk) 17:39, 11 December 2024 (UTC)

Standards from the style sheets of academic journals in Mathematics, Physic and Engineering

Since the style sheets of academic journals in mathematics, physics and engineering all agree since about 1920, I'm not sure why this is still so controversial.

I haven't seen any variance in the rules used by journals in the relevant fields, I think it is fairly clear

 Groupings (parenthesis, brackets, fraction bars)
 Unary Subtraction
 Exponents
 Juxtaposition (also called implied multiplication)
 Multiplication and Division
 Addition and Subtraction
 - when calculations are of equal precedence they are resolved from left to right
 - and the clarification that multiple exponents are read from the top down  — Preceding unsigned comment added by 2601:180:8300:8C50:DC15:E3C6:CE13:601F (talk) 21:50, 13 September 2023 (UTC) 

I would like to see the source for this. I do know that some physics journals prioritize juxtaposition but have never seen a math journal that did. There is no such operation as "unary subtraction". Subtraction is a binary operation. The unary minus is "negation". Rick Norwood (talk) 09:59, 14 September 2023 (UTC)

I would also like to see the source for this quote. My take is that if there really was an agreed upon standard we wouldn't see the variation among computer programming languages - the people who write the language specs are certainly capable of reading and applying a standard. Mr. Swordfish (talk) 17:19, 14 September 2023 (UTC)

Programming languages have different constraints than mathematical publication. In particular, the basic operators (+, -, *, /) do not obey the associative law: integer calculations can overflow depending on association, and floating-point calculations can give different results. So unlike in mathematics, how operations associate is important. Different languages also have different philosophies about reordering operations: some specify the order precisely, others allow the implementation to reorder. Again, this is not relevant to mathematics. Finally, mathematicians simply avoid writing anything ambiguous, whereas programming languages must accept any input they're given.

So I don't think you can draw conclusions about mathematical notation by looking at what programming languages do. --Macrakis (talk) 21:17, 15 September 2023 (UTC)

Hi, sorry, that 'quote' was me. I didn't intend it as a quote but as a generalization of many sources I've read. I guess I'm a noob in the Wikipedia editing system. First off, I did mean "unary minus" not "unary subtraction"; and also that line is wrong because -3^2 is -(3^2) not (-3)^2. So yes, that line is wrong or out of order. Second, I think exponents should be considered a type of grouping like fraction bars are. Third multiplication by Juxtaposition does seem to come before multiplication and division every where I check. Because 6/2n always means 6/(2n) not 3/n. 2601:180:8300:8C50:A1C5:F1DD:560E:BA72 (talk) 17:12, 18 September 2023 (UTC)

An interesting example is Physical Review Style and Notation Guide which says Multiplication *always* precedes division but also prohibits all multiplication signs except for a very special case involving line wraps inside an equation. So in this guide multiplication comes before division but all multiplication is by juxtaposition. 2601:180:8300:8C50:A1C5:F1DD:560E:BA72 (talk) 17:38, 18 September 2023 (UTC)

In physics they have their own rules. In mathematics, different rules. The only way to deal with this situation rationally is to use parentheses, e.g. 6/(2n) or (6/2)n. Rick Norwood (talk) 10:00, 19 September 2023 (UTC)

An expression such as $x/2y$ unambiguously means $x/(2y),$ and readers have no trouble interpreting this in ordinary circumstances, irrespective of whether they are in physics, mathematics, or any other field. If the other meaning were intended, it should instead be written ${\tfrac {1}{2}}xy$ or ${\tfrac {x}{2}}y$ or $(x/2)y,$ etc. –jacobolus (t) 03:18, 12 January 2024 (UTC)

I'd never call "$x/2y$" unambiguous. If I meant "$x/(2y)$", I'd prefer to write "${\tfrac {x}{2y}}$" to make that clear. If you have to write a program implementing some computation from some physics paper, and you come across "$x/2y$", you better complain the ambiguity to its author than translate it to the most similar x / 2 * y. - Jochen Burghardt (talk) 17:03, 14 January 2024 (UTC)

It was perfectly unambiguous until people started disagreeing on the interpretation, just like what happened to the words "trapezium" and "billion" (which, incidentally, all stem from the United States. What's up with that?). $x/2y$ means $x/(2y)$, and if you wanted/intended $0.5*x*y$ then explicitly write out the multiplication symbol, $x/2*y$. 203.218.11.233 (talk) 08:20, 5 February 2024 (UTC)

Is the trapezoid controversy you are talking about whether to consider a parallelogram a kind of trapezoid, or the controversy about whether "trapezoid" means the same as "trapezium" or whether it should mean a quadrilateral with no parallel sides?

The ancient Greek "exclusive" definition where a trapezia can't have more than one pair of parallel sides is a bad one IMO, comparable to the bad choice of definition that 1 (one), as a "unit", was not really a "number". –jacobolus (t) 17:12, 5 February 2024 (UTC)

I can not find the specific sentence "Multiplication *always* precedes division". Can someone help out? 62.46.182.236 (talk) 23:04, 24 February 2024 (UTC)

Misrepresentation of Source

"In academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division,[28]"

I looked at the source, and yes, it says that multiplication is of higher precedence than division. However, it does NOT say that this is only true in cases where there is implied multiplication. The phrase "for example" implies that this source should support the previous sentence, which is does not.

The other (unlinked) sources in the paragraph again support multiplication having higher precedence than division, though whether implied multiplication is relevant is unspecified. That leaves the claim with no supporting source. 50.86.240.11 (talk) 20:56, 7 February 2024 (UTC)

The only thing that is clear is that insisting that multiplication takes precedence over division, whether in some cases or in all cases, leads to endless argument and confusion. What sources say is: avoid ambiguity. In physics, the matter may be decided, but not in mathematics. And it seems to me unnecessary to have one rule for some disciplines and a different rule for other disciplines. Rick Norwood (talk) 12:02, 11 February 2024 (UTC)

I agree, multiplication is typically taken to have higher precedence than division, and this essentially never causes confusion except (a) for introductory students who are not yet used to ordinary notational conventions of written mathematics, and (b) in viral facebook images using notation that is never used in practice, aimed at bored laypeople who only vaguely remember anything they learned in school. –jacobolus (t) 22:41, 11 February 2024 (UTC)

"multiplication is typically taken to have higher precedence than division" there is no proove for that claim. Executing symbols logically it is exact the other way around. If you take physics into account, onle from left to right can be right.

https://math.ucr.edu/home/baez/physics/General/binaryOps.html 62.46.182.236 (talk) 23:24, 24 February 2024 (UTC)

I'm a mathematician. My comment about what mathematicians do and what physicists do is based on my experience, and is not something I'm trying to add to the article. The article should simply state that there are two views on the subject. Actually, three views, since some people evaluate 6/2*x differently from 6/2x.

The "left to right" rule is simply wrong, in mathematics, physics, and in everyday arithmetic. The correct rules are that addition and multiplication are commutative and associative and pure addition and pure multiplication can be done in any order. Nobody familiar with numbers is going to evaluate 5 x 765 x 2 from left to right.Rick Norwood (talk) 11:12, 25 February 2024 (UTC)

Don Koks' argument about the meaning of "1/2 second" doesn't seem fully baked to me. I would interpret "1/2 second" to probably mean (1/2) second, but in the opposite direction, I would interpret "1 meter / 2 seconds" to probably mean (1/2) meters/second, not (1/2) meters·seconds. Either of these would be improved by better typography: "${\tfrac {1}{2}}$ seconds" is entirely unambiguous. –jacobolus (t) 14:09, 25 February 2024 (UTC)

Treatment of internet memes

An editor recently removed this from the lead, stating that it was unnecessary.

My take is that a majority of the traffic to this page is a result of an argument over some ambiguous internet meme. I don't have anything to back this up, it's just a hunch.

Anyway, it seems worth discussing here on the talk page - should this one-sentence treatment be in the lead? I think it should. Other opinions? Mr. Swordfish (talk) 20:50, 11 February 2024 (UTC)

Internet memes should not be in the lead section. They are nowhere close to an essential part of understanding the topic. Including 1–2 sentences somewhere in the article body is more than sufficient.

Moreover, "knowyourmeme" etc. are not reliable sources. See WP:KNOWYOURMEME. –jacobolus (t) 22:38, 11 February 2024 (UTC)

Hi, Jacobolus. Your comment on your recent edit seems to be a reference to my most recent edit, but none of the things you deleted were caused by my most recent edit, which only changed a single word. I don't think I wrote anything you deleted, though I wouldn't swear to it. I have no objection to taking out all the references to the internet memes, though they may be of interest as a minor point later in the article. Rick Norwood (talk) 22:45, 11 February 2024 (UTC)

@Rick Norwood I think maybe you were editing from a previous version of the page and didn't de-conflict the intermediate edits? Your change special:diff/1206338161 was essentially a revert of the previous several edits. If what you were trying to do was add a link to the "resource center" tutoring webpage, I don't think that counts as a reliable source. –jacobolus (t) 23:41, 11 February 2024 (UTC)

@Rick Norwood Is it okay with you if we say that implied multiplication "typically" binds tighter than division in academic literature? I personally have never seen a counter-example (with the exception of computer code), and have surely read at least hundreds of examples of papers using notation like a / bc to mean a / (bc), across a variety of technical fields. –jacobolus (t) 01:14, 12 February 2024 (UTC)

Most of the sources I am familiar with (in pure mathematics) disagree that it is standard, pointing out numerous problems with the idea. But the important point is that it is ambiguous, and has no advantages. How much harder is it to type a/(bc) than to type the ambiguous a/bc?

Rick Norwood (talk) 02:20, 12 February 2024 (UTC)

Do you have an example of a source making this claim? Or an example of a source which implicitly uses the opposite convention that $a/bc=(a/b)c$? I see these expressions which you claim are ambiguous all the time in pure mathematics works (and computer science, and applied math, and engineering, ...), from the 18th century down to the present day, and have never once seen an example where this the intended interpretation was the other way. Thus this is not really ambiguous in practice; the convention is well established and widely understood. The advantage is that it reduces clutter, which can sometimes be tremendously helpful. –jacobolus (t) 02:34, 12 February 2024 (UTC)

I'll try to add more context to this section. Still skimming sources. –jacobolus (t) 06:14, 12 February 2024 (UTC)

Okay, I've expanded those sections a bit, added more sources, and taken out most of the questionable self-published sources. Sorry for the history spam, folks: when I reread paragraphs I second-guess the previous wording, and end up making repeated passes of minor changes. –jacobolus (t) 20:51, 12 February 2024 (UTC)

I've been thinking about this quite a bit. Your rewrite has improved the article greatly, and as it stands, I have no strong objection. The bigger problem is that every math book used in K-12 education in the United States lies to its students. For example, they all say that parentheses are an "operation", just like addition and multiplication. And they all say that you must do parentheses first, which is impossible in a problem such as 2+3+(x+y). And they all say you must work from left to right, which is ridiculous in a problem like 283+389-283.

However, getting back to the question at hand. As you not, the problem only occurs with the use of the solidus. I've just glanced through several math books, and they almost always use a horizontal fraction like. I haven't found one that uses x/2 instead of 1⁄2x or x⁄2.Rick Norwood (talk) 13:01, 14 February 2024 (UTC)

Oops. I could not get Wikipedia to use a horizontal fraction bar. But in the three examples I gave, even thought they use a solidus, they use the solidus in a way such that there is no ambiguity.Rick Norwood (talk) 13:03, 14 February 2024 (UTC)

(You can get a vertically stacked "inline" size fraction using <math>\tfrac12 x</math> which renders as ${\tfrac {1}{2}}x$ or using {{math|{{sfrac|1|2}}''x''}} which renders as ⁠1/2⁠x.)

All of the forms $x/2,$ ${\tfrac {x}{2}},$ and ${\tfrac {1}{2}}x$ are quite common in books and papers in pure math, in contexts where a full-sized fraction wouldn't fit or where vertical space is at a premium; this includes not only "inline" equations in running prose, but also within "display" style equations in nested fractions, superscripts, limits of sums, etc. One of the results that popped up in a web search about order of operations was a quora or stackexchange discussion (can't remember which) in which one participant did some examination of several papers by Fields Medalists, and found multiple examples of fractions like $a/bc$ meaning $a/(bc).$ In my experience this convention is an unremarkable feature of mathematical writing, and is not confusing in practice. –jacobolus (t) 15:40, 14 February 2024 (UTC)

As to your comment about ~5th–8th grade textbooks: you are right that they are typically misleading about this topic. The issue is that mathematicians use notation as a form of communication, whereas middle school textbooks use mathematical notation as a set of prescriptivist rules. The rules established by someone trying to make something very precisely specified don't necessarily match the practical usage of a community of writers. This is similar to the problem of setting down prescriptivist "grammar rules" and teaching them to students; many such rules are routinely violated in professional writing. –jacobolus (t) 15:46, 14 February 2024 (UTC)

@Mr swordfish I've made a bunch of other changes relevant to the ambiguity of multiplication/division, internet memes about it, and related topics. Does the current version address your concern, or do you still think the memes are under-discussed? @D.Lazard, @Jochen Burghardt do these recent changes seem okay to you, or are there parts that seem problematic? –jacobolus (t) 02:11, 17 February 2024 (UTC)

Thanks for asking. Meanwhile, I've lost overview, but it seems your edits were fine. - Jochen Burghardt (talk) 19:07, 17 February 2024 (UTC)

I think the material that is currently there is very good and your edits are an improvement - including the quote from Hung-Hsi Wu in particular.

Here's my take: When I edit articles on Wikipedia, I try to keep the likely audience in mind. Of course, I don't have any audience research data to go by so my idea of the likely audience may by off base, but then nobody else has that research data either so we need to respect others' opinions if they are different than ours. For this article, I don't think the typical reader is a mathematician, scientist, or engineer. I do think that a significant percentage of our visitors are here to answer the question "What is the answer to that stupid math formula on facebook?" If I am correct about this, then as a service to our audience we should make it easy to find that answer.

One way to make that easy was to include a single sentence in the lede. I'm sure that there are other ways. Right now, it's somewhat buried as the last paragraph of the second subsection of the second section and my preference would be to make it easier for the readers to find it. I'm open to other ways to make it more easily discoverable, but a single short sentence in the lede seems to be the simplest way to address my concern. Mr. Swordfish (talk) 21:46, 17 February 2024 (UTC)

Personally I think that would be "undue weight" in an article about order of operations. But plausibly this facebook meme could be its own article (there are several reliable sources discussing it) if you really think it would be helpful to people. –jacobolus (t) 22:37, 17 February 2024 (UTC)

I don't think it's sufficiently notable to have it's own article, and I don't know that there's much more to say about it than the current paragraph so the article would probably permanently remain a stub. I wouldn't object is someone created it, but I wouldn't advocate for it. And then there's the practical problem of how to title such an article so that people looking for it can find it - I can't think of one.

As for undue weight, from what I've seen the only people discussing this topic on line (other than here at this talk page) are the ones arguing about "that stupid math problem on facebook". Mr. Swordfish (talk) 23:14, 17 February 2024 (UTC)

In my opinion Wikipedia shouldn't decide on how to organize or fill articles based on what people discuss on social media. YMMV. –jacobolus (t) 00:06, 18 February 2024 (UTC)

I can respect that opinion, but it's orthogonal to the question of undue weight which is what I was responding to.

My take is that we should serve the audience as opposed to creating the platonic ideal of the perfect article. Mr. Swordfish (talk) 00:12, 20 February 2024 (UTC)

To quote WP:UNDUE, "Keep in mind that, in determining proper weight, we consider a viewpoint's prevalence in reliable sources, not its prevalence among Wikipedia editors or the general public." I think mentioning this topic at all is entirely sufficient, and promoting it to the lead doesn't seem justified to me. Maybe we should take the question to a more visible venue like WT:WPM for more feedback, if you think this seems like a controversial position. –jacobolus (t) 05:40, 20 February 2024 (UTC)

Point taken about WP:Undue. Been a while since I read it.

As for taking it to Wiki Project Mathematics, I have no objections but I'm also satisfied if we settle it here on this talk page. So far my concern seems to have been met with a MEH? and if that's the case so be it. If anyone else wants to weigh in I'm sure they know how. Mr. Swordfish (talk) 22:27, 20 February 2024 (UTC)

@D.Lazard, @Jochen Burghardt – any thoughts on including a sentence about facebook memes in the lead section? –jacobolus (t) 07:33, 21 February 2024 (UTC)

Another thing that might be helpful is more images. A picture of such a meme directly might help readers find the relevant discussion (though this might be gratuitously distracting).

Another type of image that would be nice would be a diagram showing the relation between a mathematical expression and a generated expression tree, maybe even a simple and a more complicated example could be pictured. –jacobolus (t) 22:46, 17 February 2024 (UTC)

ISO 80000

Should we be including ISO standards in the "Mixed division and multiplication"? The standards include authoritative answers to some of the questions and ambiguities, for instance 80000-2-(9.6) states that '÷' "should not be used" for division (see division sign) and 80000-1 (7.1.3) states that the solidus "shall not be followed by a multiplication sign or a division sign on the same line unless parentheses are inserted to avoid any ambiguity". Unfortunately the standards aren't freely available and I have only come across snippets that others have posted elsewhere. StuartH (talk) 05:48, 10 April 2024 (UTC)

Seems fine to mention, though I'm not sure anyone follows this per se, in practice. –jacobolus (t) 06:12, 10 April 2024 (UTC)

I've added as a minor update for now - I think you're right that very few people even know about the standard but it is still the standard and probably warrants a mention. StuartH (talk) 09:24, 10 April 2024 (UTC)

Picture

I like the idea of a picture at the top of the page, and I even like the picture. But, sadly, it seems much too complicated for readers who are not mathematicians. Rick Norwood (talk) 10:33, 16 July 2024 (UTC)

Maybe something like this?

http://sweeneymath.blogspot.com/2011/05/how-i-see-exponent-rules-and-log-rules.html

Rick Norwood (talk) 10:40, 16 July 2024 (UTC)

I hadn't noticed the picture. I agree it could be better. A couple of possibilities I can imagine are (1) relation of a (not too) complicated expression to a tree (cf. binary expression tree, parse tree) which is effectively what the order of operations describes, (2) a [slow] animation showing evaluation of a numerical example from inside outward. –jacobolus (t) 16:18, 24 August 2024 (UTC)

Syntax trees of expressions can be found at commons:Category:Syntax_trees, e.g. commons:Exp-tree-ex-11.svg. Imo, any animation [no matter at which speed] severly distracts a reader's attention - so, while it is a good idea to provide an animation as you described, it is a bad idea to let it run within the article. Instead, it could be put in the category, and its name could be linked from the article. - Jochen Burghardt (talk) 17:16, 25 August 2024 (UTC)

One concern with a tree is that it might be confusing to some of the intended audience. –jacobolus (t) 17:41, 25 August 2024 (UTC)

You have a point there. What about modifying the current image such that in each line, one subexpression is evaluated? We'd need to replace "a" by some number for this (*); and probably we'd start from a less involved expression. If the changed parts are highlighted, an impression of a tree-like structure will arise (somewhat like in the right part of the bottommost picture), but without the need to talk about the concept of syntax tree.

(*) BTW: Maybe, we should mention somewhere in the article that only ground expressions can be evaluated, independent of the picture issue? - Jochen Burghardt (talk) 18:19, 25 August 2024 (UTC)

history section

@Gronk Oz – can you find better sources for the claims you are making here? One of your sources is a (non-expert) university professor's personal website repeating the claims of your other link, an email from Dave Peterson, a former software engineer and community college teacher who was part of the "Ask Dr. Math" team, which is now defunct but with some of the same people at the website themathdoctors.org. None of these is a peer reviewed source, or really cites its sources, and while I think Ask Dr. Math / The Math Doctors was/is a nice website, it doesn't really meet Wikipedia's "reliable sources" standard. I don't think this history seems quite right, which is unsurprising for an informal email reply from a hobbyist (as compared to e.g. a professional historian doing careful research and publishing in a peer-reviewed journal). –jacobolus (t) 19:12, 4 October 2024 (UTC)

@Jacobolus: I agree these are not great sources. I thought the absence of a History section was a real deficiency in this article, so I wanted to get the ball rolling with what sources I could find. Now that I look into it further, there is what looks like a good resource at web.archive.org/web/20020621160940/http://members.aol.com/jeff570/operation.html - while it is still a blog-style entry, it refers to a number of published works that would be worth following up - especially A History of Mathematical Notations (1928-1929) by Florian Cajori. Unfortunately, I don't have access to those, so I will keep looking.--Gronk Oz (talk) 03:00, 5 October 2024 (UTC)

I have found an online archive of the Cajori book at archive.org/details/historyofmathema031756mbp/mode/2up. But it will take some time until I can address this, since I am caught up with real-life matters at the moment. I will try to get to it when I can, unless somebody else wants to... --Gronk Oz (talk) 03:07, 5 October 2024 (UTC)

The relevant part of Jeff Miller's site (one of the best sources on the web about the history of mathematical terms) is now at https://mathshistory.st-andrews.ac.uk/Miller/ – this page specifically is https://mathshistory.st-andrews.ac.uk/Miller/mathsym/operation/ –jacobolus (t) 04:07, 5 October 2024 (UTC)